The class blog for Math 3010, fall 2014, at the University of Utah

Mathematical Psychology

What if I told you I could explain every single one of your decisions and actions using math? WHAT? But humans are so unpredictable and random! We really don’t understand these strange creatures at all.

Enter mathematical psychology:

Ernst Weber. Image: Public domain, via WIkimedia Commons.

The Society for Mathematical Psychology says that mathematical psychology is “broadly defined to include work of a theoretical character that uses mathematical methods, formal logic, or computer simulation.” What does that mean? It basically means that mathematical psychology can be any type of psychological work that uses math to explain something or figure something out.

Gustav Fechner. Image: Public domain, via Wikimedia Commons.

The first traces of mathematical psychology go way back to Ernst Weber and Gustav Fechner in the 19th century. For the first time in history, these guys applied mathematical techniques to psychological processes. The moment they stepped out from behind the desk and got their hands busy, both mathematical psychology and experimental psychology were born. These twins went hand in hand toward making psychology a quantified science.

Around the same time, researchers in astronomy were mapping the distance between stars by jotting down the exact time a specific star passed through the cross-hairs on their telescope. There were no automatic instruments, so the measurements were based on human response speed. The German astronomer Friedrich Bessel (who never attended a university) studied the differences between the measurements. Based on measurements of general human response speed, he used math to create personalized equations for each of his fellow astronomers. The equations would actually cancel out the personal differences from astronomer to astronomer.

Completely independently, the physicist Hermann von Helmholtz measured people’s reaction times to determine the speed of nerve conduction. Most of Helmholtz’s colleagues though that nerve signals passed along nerves immeasurably fast. Helmholtz used a freshly dissected sciatic nerve and calf muscle from a frog and attached an altered galvanometer as a timing device. He reported that the transmission speed is in the rage of 24.6 – 38.4 meters per second. Today we know that nerve impulses travel anywhere between 1 and 120 meters per second depending on the diameter of the fiber and the presence or absence of myelin, so Helmholtz was pretty on point.

The Dutch physiologist F.C. Donders and his assistant J. J. de Jaager combined these two branches of research to use the reaction times to figure out objectively the amount of time that it takes to complete elementary mental operations. He came to the conclusion that simple processes (automatic processes) are much faster than choice processes(things we decide to do), but his numbers were pretty far off due to lack of technology. However, a German psychologist named Wilhelm Wundt used Donder’s ideas to create the first psychological laboratory. It was hard to replicate the results from the lab, because of his method of introspection. In other words, he was observing his own results, so his own individual differences changed the results, but the lab was a huge turning point in the Mathematical Psychology.

Wilhelm Wundt in the first psychological laboratory. Image: Public domain, via Wikimedia Commons.

After the introspection method didn’t work out, two different schools of thought developed. People either gravitated towards studying the general conscious human experience or they were drawn to study the individual differences between humans.

Throughout the 20th century, the idea of behaviorism popped up in America and shifted the focus away from reaction times and more towards learning theory. Learning theory attempts to study the way people learn through objectively measuring their behavior (think Pavlov’s Dog). Throughout the Second World War, the military brought together a smorgasbord of mathematicians, engineers, physicists, economists and experimental psychologists in order to learn about human performance and limitations. These men and women essentially shaped mathematical psychology into what it is today.

In 1951, two guys named Bush and Mosteller published a paper titled “A Mathematical Model for Simple Learning” which presented the first detailed data on learning experiments. From there, people began to publish more papers, teach courses, compile the mathematical models into volumes, create a Journal of Mathematical Psychology and otherwise take mathematical psychology seriously. Today mathematical psychology is a tried and true branch of psychology that many researchers could not work without. We’re taking baby steps toward understanding the brain one equation at a time.